Testing the broad applicability of the PBEint GGA functional and its one-parameter hybrid form

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by  E. Fabiano

Testing the broad applicability of the PBEint GGA functional and its one-parameter
hybrid form
E. Fabiano,1 Lucian A. Constantin,2 and F. Della Sala1,2
1National Nanotechnology Laboratory (NNL), Istituto Nanoscienze-CNR, via per Arnesano 16, I-73100 Lecce, Italy
2Center for Biomolecular Nanotechnologies @UNILE,
Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy
(Dated: January 25, 2013)
We review the performance of the PBEint GGA functional (Phys. Rev. B 2010, 82, 113104) re-
3
centlyproposedtoimprovethedescriptionofhybridinterfaces,andweintroduceitsone-parameter
1
0 hybridform (hPBEint). Weconsiderdifferent well established benchmarksfor energetic and struc-
2 turalpropertiesofmolecularandsolid-statesystemsaswellasmodelsystemsandnewlydeveloped
benchmark sets for dipole moments and metal-molecule interactions. We find that PBEint and
n
hPBEint (with 16.67% Hartree-Fock exchange) yield the overall best performance, working well
a
for most of the considered properties and systems and showing a balanced behavior for different
J
problems. In particular, due to their well-balanced accuracy, they perform well for the description
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of hybridmetal-molecule interfaces.
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PACSnumbers: 71.10.Ca,71.15.Mb,71.45.Gm
]
h
p
- INTRODUCTION [13],RPBE [14], BLYP [15, 16], OLYP [16, 17]) or solids
m (PBEsol [10], ARPA+ [18], AM05 [19], Wu-Cohen [20])
e Density functional theory [1] (DFT) is nowadays one are obtained. On the other hand, hybrid functionals
h relystronglyontheunderlyingGGAapproximationsand
of the most popular computationalmethods in quantum
c
sharemostoftheir fundamentaldrawbacks. Inaddition,
. chemistry and solid-state physics. Despite its success
s they might suffer from a certain degree of empiricism
c however, DFT still suffers some limitations due the ap-
i proximationsusedintheexchange-correlation(XC)func- used to construct the partial inclusion of the Hartree-
s
Fock exchange. Thus, hybrid functionals generally im-
y tional,whichdescribesallthe quantumelectron-electron
h interactions. In fact, the development and optimization prove over the GGA description but cannot provide a
p really homogeneous level of accuracy for a wide range of
ofadvancedXCfunctionalsiscurrentlyaveryactivefield
[ applications, showing occasionally important failures for
of research[2].
1 In the years numerous different XC functionals have specific problems(e.g. transition-metalchemistry[21]or
v metal clusters [22]).
been developed, ranging from local density approxima-
8
tions [3] (LDA) to the most advanced orbital-dependent Recently,agrowingeffortwasdevotedtowardsthede-
0
8 XC functionals [4–8] including exact exchange and se- velopment of GGA functionals that could provide a well
5 lected exact-correlation contributions. The largest pop- balanced description of a large number of properties. In
1. ularity in practical applications is however own by two fact,awiseselectionofimportantexactconstrainsofthe
0 classes of functionals: generalized gradient approxima- XC energy(ase.g. inPBEorAPBE)canbe usedto op-
3 tions (GGA) and hybrid functionals. The former are in timize the performance of GGA functionals in different
1 fact, the method of choice for the investigation of large contexts [23]. Two noteworthyexamples are the PBEint
:
v systems (e.g. in biochemistry or solid-state physics) due [24] and the HTBS [25] functionals. These functionals
i to their very favorable cost-to-accuracy ratio, while the are not only equally accurate for molecular and solid-
X
latter are the workhorse for computational chemistry. state energetic andstructuralproperties,but potentially
r
a Moreover, the GGA functionals attract theoretical in- important for interface physics, surface science and clus-
terest because they are the building blocks of the more ter chemistry. Furthermore, they constitute an optimal
advanced meta-GGA, hybrid, and hyper-GGA function- starting point for the construction of hybrid functionals
als. of broad applicability.
Due to their simple form GGA functionals cannot ful- In particular, the PBEint functional has been devel-
fillallthe exactconstraintsofthe XCenergyandcannot oped to respect different exact constraints of the XC en-
be accurate for both atoms and solids [9, 10]. There- ergy and to connect the slowly-varying density regime,
fore, some selective criteria must be employed for their where the second-order gradient expansion (GE2) of the
construction. Popular approaches are to develop rather exchange energy is correct, and rapidly-varying density
specializedfunctionalsbyfittingtospecificproblemsand regime, where the PBE behavior is accurate. Thus, it
training sets or by requiring the satisfaction of selected is reasonably accurate for an important set of molecular
exact constraints of the XC energy. As a result, GGA andsolid-solidstatepropertiesand,thankstoitswellbal-
functionals formolecules(PBE[11],APBE[12],revPBE ancedbehavior,especiallysuitedforinterface[24],metal-
2
cluster[22]andsurface[26]problems. Werecallalsothat ansatz
the PBEintXC hole density shows exact properties [26],
beyond the PBE and PBEsol ones [27], related to the αs2
µ(s)=µGE2+(µPBE−µGE2) , (4)
accurate analysis of metallic surfaces [28]. 1+αs2
Inthispaperweproposeashortreviewandamorede-
tailed assessment of the performance of PBEint for dif- with α = 0.197. Equation (4) assures that the following
ferent problems in chemistry and physics. In addition, minimalconstraintsaresatisfied: i)intheslowly-varying
we use the PBEint functional to build a global hybrid densitylimitµ→µGE2;ii)intherapidly-varyingdensity
functional and investigate the results of this for differ- limit µ→µPBE; (iii) the fourth-orderterm(∝s4) in the
ent amounts of the Hartree-Fock (HF) exchange mixing. exchange gradient expansion vanishes.
We find that a rather small amount of HF mixing (only Due to Eq. (4) the exchange enhancement factor of
16.67%)givesthebestresultsoverabroadrangeofprop- PBEintvariessmoothlybetweenthatofPBEsol(atsmall
erties. We recall that global PBE and PBEsol hybrids s values) and that of PBE (at larger s values), granting
are instead commonly constructed using 25% and 60% a good description of all possible density regimes and of
of HF mixing [29], respectively. Thus, the PBEint ex- many different systems from molecules to solids. Note
change construction is more compatible with the exact however that the PBEint functional (unlike HTBS) is
exchange,a result thatwas alreadyprovedin the caseof notasimpleinterpolationbetweenPBEandPBEsol,but
jellium surfaces [26]. providesinsteadaphysicallymeaningful(althoughinter-
polated) value for the µ parameter at different density-
regimes. In fact, a similar construction proved to be
THEORY AND COMPUTATIONAL DETAILS useful also in the case of noninteracting kinetic energy
functionals [33].
The PBEint functional is constructed considering an Forthecorrelationpart,thePBEintfunctionalutilizes
exchange functional again a PBE-like expression [11]
E = ρ(r)ǫunif(ρ(r))F (s(r))dr, (1) E = ρ(r) ǫunif(r (r),ζ(r))+H(r (r),ζ(r),t(r)) dr,
x Z x x c Z c s s
(cid:2) (cid:3)
(5)
with a PBE-like enhancement factor [11] with
κ
Fx(s)=1+κ− 1+µs2/κ, (2) H(rs,ζ,t)=γφ3ln(cid:18)1+ βγ 1+tA2t+2A+tA42t4(cid:19), (6)
where ρ is the electron density, ǫunif =
x where ρ= ρ +ρ is the total density, r =[(4π/3)ρ]1/3
−(3/4)(3/π)1/3ρ1/3istheexchangeenergyperparticleof ↑ ↓ s
is the local Seitz radius, ζ = (ρ −ρ )/ρ is the relative
the uniform electron gas, and s = |∇ρ|/(2(3π2)1/3ρ4/3) ↑ ↓
spinpolarization,φ=[(1+ζ)2/3+(1−ζ)2/3]/2 isa spin
is the reduced gradient for the exchange. In Eq. (2) the
scaling factor, ǫunif is the correlation energy per particle
value of the parameter κ can be easily fixed to 0.804 c
oftheuniformelectrongas,Aisafunctionofǫunif andφ
by imposing that the Lieb-Oxford bound [30] for the c
(seeRef. 11), t=(3π)1/6|∇ρ|/(4φρ7/6)isthecorrelation
exchange energy holds locally [11] (i.e. for the exchange
reduced gradient, and γ = (1 − ln2)/π2 is a parame-
energy density at each point in space). The value of the
ter fixed by uniform scaling to the high-density limit of
µ parameter can be instead fixed by noting that Eq. (1)
the (spin-unpolarized) correlation energy. The parame-
has, through second-order,the gradient expansion
ter β, which is fixed to 0.066725 in the original PBE by
the second-order gradient expansion of the correlation
E(GE2) = ρ(r)ǫunif(ρ(r))[1+µs2]dr , (3)
x Z x energy,isobtainedforPBEintbyfittingtoajelliumsur-
face reference system: β =βPBEint = 0.052. In fact, the
so that µ can be just identified with the second-order PBEintcorrelationcanreproducewithhighaccuracythe
coefficientof the exchangegradientexpansion. However, wave-vector analysis of the correlation surface energies
no unique value can be fixed for µ with this condition. ofjellium slabs of different thickness and r [26], outper-
s
In fact, in the slowly-varying density limit (s → 0) µ = forming other PBE-like GGA functionals. This result is
µGE2 = 10/81 [31], while in the rapidly-varying density important to prove the ability of PBEint to describe ac-
limit (s ≫ 1) µ = µPBE = 0.2195 was found to be a curatelysystems where differentdensity regimes coexist.
good choice. Finally, from the semiclassical-atomtheory The value of βPBEint is also intermediate between those
itwasfoundµ=µMGE2 =0.26[12,32]. ForthePBEint ofPBEandPBEsolassuringanaccuratecompromisefor
functionalwe thereforeabandonthe idea ofa constantµ a large palette of systems ranging from atoms to solids
andconsiderinsteadans-dependentµwiththefollowing [26].
3
Hybrid functional light basis-set and a 18×18×18 k-point grid. In this
case, scalar relativistic effects were accounted for by the
To constructour hybridfunctional we follow the adia- zeroth-order relativistic approximation (ZORA) [44].
baticconnectionschemeintroducedinRef. 34wherethe Inmoredetails,inthe presentworkthefollowingtests
XC (hybrid) functional is obtained as the result of the were considered:
coupling-constant integration
AE6: AtomizationenergiesofSiH ,SiO,S ,C H ,
4 2 3 4
1
C H O ,andC H ;referencedataweretakenfrom
Ehyb = Ehyb(λ)dλ, (7) 2 2 2 4 8
xc Z xc Ref. 45. Note that the AE6 test set was build
0
to be representative for the results of the large
andthefollowingansatzisusedforthehybrid-functional
Database/3 [46], including 109 atomization ener-
coupling-constant decomposition
gies.
Ehyb(λ)=EGGA(λ)+(EHF−EGGA)(1−λ)n−1, (8)
xc xc x x
W4: Atomization energies from the W4-08woMR
withEHFbeingtheHartree-Fockexchangeenergy,EGGA
x x test set of Ref. [47]. It includes 83 atomizationen-
being the exchange energy of a given GGA functional,
ergies of organic molecules selected from the origi-
and EGGA(λ) its coupling-constant decomposition. The
xc nalW4testset[48]excludingmulti-referencecases.
parameter n controls the balance between the nonlocal
Hartree-Fock and the local GGA exchange at different TM10: Atomization energies of CrH, MnH, CoH,
values of the coupling constant and is related to pertur- V ,Sc ,TiO,MnO,CuO,CrF,andCuF.Reference
2 2
bation theory considerations [34]. data were taken from Ref. 21.
If in Eq. (8) we use GGA=PBE and n=4, after per-
forming the integration (7), the PBE0 [35] functional AUnAE: Atomization energies of the Au−2, Au2,
(also known as PBE1PBE) is obtained. In this work Au3, and Au5 clusters. Reference data, includ-
instead we consider GGA=PBEint and we end up with ing relativisticandthermalcorrections,weretaken
the hPBEint functional from Ref. 22.
1
EhPBEint =EPBEint+ (EHF−EPBEint) . (9) K9: Barrier heights and reaction ener-
xc xc n x x
gies of three organic reactions. Namely,
In this case we do not fix the value of the parameter n,
OH+CH →CH +H O, H+OH→O+H , and
4 3 2 2
but we will consider n= 4 ,5, and 6 in order to assess
H+H S→H +HS. Reference data were obtained
2 2
the optimal value of this parameter on a broad range of
from Refs. 45, 49.
situations. In fact, using a value of n = 4 can be better
to describe molecularsystems(as in PBE0),while larger HB6: Bindingenergiesofthehydrogen-bondinter-
values of the parameter might be more appropriate for actingsystems (H O) ,(HF) , (NH ) , NH –H O,
2 2 2 3 2 3 2
transition-metals, clusters and interfaces. (HCONH ) ,and(HCOOH) . Referencedatawere
2 2 2
taken from Ref. 50.
Computational details DI6: Binding energies of the dipole-dipole inter-
actingsystemsCH Cl–HCl,CH SH–HCl,CH SH–
3 3 3
We tested the PBEint and hPBEint functionals for a NCH, (H2S)2, (HCl)2, and H2S-HCl. Reference
series of molecular and solid state properties including data were taken from Ref. 50.
atomization energies, structural properties, non-bonded
MGBL19: Bond lengths of H , CH , NH , H O,
interactionsandreactionenergies. Thecalculationswere 2 4 3 2
HF, C H , HCN, H CO, OH, CO, N , F , CO ,
performed with the PBEint functional as well as with 2 2 2 2 2 2
N O, and Cl . Reference values were taken from
the hPBEint functional with n=4, 5, and 6. Moreover, 2 2
Ref. 51. Note that this set provides a global as-
calculations employing the PBE [11], PBEsol [10], and
sessment over bond lengths involving one hydro-
PBE0 [35] functionals were also carried out for compar-
genatomandbondlengthsnotinvolvinghydrogen
ison. In all calculations a def2-TZVPP basis set [36, 37]
atoms, which usually behave differently for differ-
was employed. For transition metals the core electrons
ent functionals [23].
were replaced with effective core potentials (ECP) [38–
40]. All calculations were performed at the optimized AUnBL: Bond lengths of Au−, Au , Au+, Au−,
2 2 3 3
ground-state geometry, except for those on the DM25
Au , Au (capped pentagon, C symmetry),
4 6 5v
and small interfaces test sets (see below).
SeAu , and (ClAuPH ) . Reference data were
2 3 2
All calculations on molecular species were performed
taken from Ref. 22.
with a development version of the TURBOMOLE pro-
grampackage [41]. Calculations on bulk solids were per- F38: HarmonicvibrationalfrequenciesofH ,CH ,
2 4
formed with the FHI-AIMS program [42, 43] using the NH , H O, HF, CO, N , F , C H , HCN, H CO,
3 2 2 2 2 2 2
4
TABLE I: Mean absolute errors for different tests as obtained from PBE, PBEint, PBEsol, PBE0, and hPBEint calculations
with n=4, 5, 6. For each line the best result is indicated by bold face, the worst is underlined . All energies are in kcal/mol
(except for COH6). All bond lengths are in m˚A.Vibrational frequencies are in cm−1.
Test PBE PBEint PBEsol PBE0 hPBEint
n=4 n=5 n=6
Organic molecules
AE6 14.5 24.7 34.9 5.4 12.4 14.3 15.7
W4 10.7 15.5 21.5 5.7 6.3 7.5 8.5
OMRE 6.8 8.2 12.0 9.1 12.8 11.7 11.1
DC9 10.6 15.5 17.6 10.2 13.6 13.9 14.2
K9 7.51 9.09 10.59 3.96 4.90 5.62 6.17
MGBL19 9 10 10 6 9 8 7
F38 56.8 65.4 65.9 51.7 53.2 45.3 45.6
Transition metals
TM10 13.4 15.5 18.3 18.4 16.2 15.3 12.0
AUnAEa 0.3 2.2 4.4 3.4 2.5 1.8 1.3
TMRE 3.7 6.9 9.9 11.1 11.1 9.4 8.3
AUnBL 78 31 25 86 58 56 55
Non-bondedinteractions
HB6 0.4 0.5 1.7 0.5 0.7 0.7 0.6
DI6 0.4 0.4 1.0 0.4 0.4 0.4 0.4
Solid-state properties
LC6 59 32 22 - - - -
COH6b 0.15 0.14 0.21 - - - -
a) atomization energy peratom.
b) eV/atom.
CO ,N O,Cl ,andOH.Referencedataweretaken (ionic solid). Reference data weretakenfromRefs.
2 2 2
from Ref. 52 54, 55. The LC6 and COH6 tests were not per-
formed for hybrid functionals.
OMRE:Reactionenergiesoforganicmolecules. It
includesCH2+H2 →CH4,F2+H2 →2HF,C6H6+ DM25: Dipolemomentsofmolecules. Thetestset
3H2 → C6H12, CO+3H2 → CH4 +H2O, SO2 + considers a broad range of systems and of dipole
3F2 → SF6 + O2, and C4H6 + C2H4 → C6H10. moments,rangingfrom0.11to11.56Debye. Inde-
Reference data were taken from Ref. 53 tail,itincludesCO,CFCl ,furan,OCS,BF,CF O,
3 2
phenol, HCOOH, CH SH, HNCO, CH COOH,
TMRE: Reaction energies of transition-metal 3 3
CH OCHO,HF,H O,CH Cl,CH ONO,pyridine,
complexes. It includes Ni(CO) +CO→Ni(CO) , 3 2 3 3
3 4 H2CO,ClCN,CuH,HCN,CHOCH OH,CH NO ,
Fe(CO) +CO→Fe(CO) , 1Cl +CoCl →CoCl , 2 3 2
4 5 2 2 2 3 LiCl, and NH (CH=CH) NO (denoted as N6).
and 2FeCl → Fe Cl . Reference data were taken 2 6 2
2 2 4 Reference data and geometries of BF, H O, CuH,
from Ref. 21. 2
H CO, LiCl, and N6 are taken from Ref. 56. The
2
DC9: Nine reaction energies of medium-size remaining reference data and (experimental) ge-
molecules,whichareusuallytreatedpoorlybyDFT ometries are taken from Ref. 57. All structures
methods. Reference values were taken from Ref. are available in supporting information [58].
47.
Small interfaces: Interaction energies of twelve
LC6 and COH6: Lattice constants and cohesive small metal-molecule systems, representative of
energiesofbulkNa(simple metal),Ag,Cu(transi- metal-molecule interfaces. The set includes Au –
2
tionmetals),Si,GaAs(semiconductors),andNaCl SH, Au+–N , Au –SHCH , Au –SH, Au+–N ,
2 2 2 3 3 3 2
5
TABLE II: Dipole moments (Debye) for different test molecules as computed with different DFT methods. The mean error
(ME), mean absolute error (MAE) and mean absolute relative error (MARE) of each method with respect to reference values
are reported in thelast lines. Foreach row the best result is highlighted in bold style, theworst one is underlined.
System PBE PBEint PBEsol PBE0 hPBEint Ref.
n=4 n=5 n=6
CO 0.22 0.24 0.23 0.10 0.11 0.14 0.15 0.11
CFCl3 0.35 0.33 0.35 0.42 0.41 0.40 0.39 0.45
furan 0.59 0.58 0.59 0.66 0.65 0.63 0.62 0.66
OCS 0.75 0.74 0.77 0.80 0.79 0.78 0.77 0.71
BF 1.02 1.04 1.03 0.94 0.95 0.97 0.98 0.79
CF2O 1.12 1.13 1.12 1.12 1.13 1.13 1.13 0.95
phenol 1.25 1.26 1.26 1.28 1.29 1.28 1.28 1.22
HCOOH 1.42 1.42 1.42 1.47 1.47 1.46 1.45 1.41
CH3SH 1.58 1.60 1.61 1.62 1.64 1.63 1.62 1.52
HNCO 1.99 1.99 2.01 2.07 2.07 2.06 2.05 1.61
CH3COOH 1.72 1.72 1.73 1.79 1.79 1.77 1.77 1.70
CH3OCHO 1.86 1.86 1.87 1.89 1.89 1.88 1.88 1.77
HF 1.78 1.79 1.80 1.83 1.84 1.83 1.82 1.82
H2O 1.87 1.88 1.89 1.92 1.93 1.92 1.91 1.85
CH3Cl 1.82 1.83 1.84 1.89 1.90 1.89 1.88 1.87
CH3ONO 2.26 2.27 2.30 2.21 2.21 2.23 2.23 2.05
pyridine 2.22 2.23 2.24 2.27 2.27 2.26 2.26 2.19
H2CO 2.23 2.24 2.25 2.42 2.42 2.38 2.36 2.39
ClCN 2.96 2.96 2.98 2.99 2.99 2.99 2.98 2.82
CuH 2.31 2.33 2.28 2.85 2.87 2.76 2.69 2.97
HCN 2.93 2.94 2.95 3.03 3.04 3.02 3.01 2.98
CHOCH2OH 2.40 2.40 2.43 2.57 2.57 2.54 2.51 2.73
CH3NO2 3.41 3.41 3.43 3.61 3.61 3.57 3.54 3.46
LiCl 6.90 6.89 6.90 7.06 7.06 7.03 7.01 7.23
N6 16.94 16.95 17.03 15.74 15.77 15.98 16.14 11.56
ME 0.20 0.21 0.22 0.23 0.23 0.23 0.22
MAE 0.35 0.35 0.36 0.27 0.27 0.28 0.29
MARE 13.5% 14.4% 14.4 8.0% 7.9% 8.9% 9.8%
Std. Dev. 1.10 1.10 1.11 0.83 0.84 0.88 0.92
Au –SCH , Au –SH, Au+–N , and Au –SHCH . hydrogenatomwithfractionalspin. Additionalde-
3 3 4 4 2 4 3
The general structure of the molecule-cluster sys- tails are provided in the corresponding subsection
tems was obtained from Refs. 59–61. Each later on.
structure was then reoptimized at the TPSS/def2-
TZVPP [36, 37, 62] level of theory. This geometry
was successively employed in all the calculations. All tests are necessarily not exhaustive, because for
ReferencevalueswereobtainedfromCCSD(T)cal- computational reasons they only include a relatively
culations[63–65]extrapolatedtothecompletebasis small number of systems. Therefore, for specific cases
set limit [66]. All structures are available in sup- they mightbe notfully representativeofthe true perfor-
porting information [58]. mance of the functionals. Nevertheless, since each test
set was constructed to have good representativity and
Model systems: We consider four sets of model because the selected tests covera broadrangeofproper-
systems for which exact or extremely accurate so- ties and systems, we believe that the present assessment
lutions are available. Namely, we consider: jellium offersafairoverviewoftheperformanceoftheconsidered
surfaces, jellium clusters, Hooke’s atoms, and the functionals.
6
TABLE III: Interaction energies (eV) of small organic molecules and gold clusters, computed with different DFT methods.
The mean error (ME), mean absolute error (MAE) and mean absolute relative error (MARE) of each method with respect to
CCSD(T) are reported in the last lines. For each row the result in best agreement with CCSD(T) calculations is highlighted
in bold style, the onein worst agreement is underlined.
System PBE PBEint PBEsol PBE0 hPBEint CCSD(T)
n=4 n=5 n=6
Au2–SH 1.99 2.20 2.35 1.57 1.72 1.81 1.87 1.98
Au+2–N2 0.99 1.13 1.27 0.84 0.95 0.99 1.01 1.30
Au2–SHCH3 0.88 1.04 1.17 0.78 0.90 0.92 0.94 1.24
Au3–SH 3.02 3.20 3.33 2.96 3.09 3.11 3.12 3.15
Au3–N2 0.77 0.93 1.06 0.67 0.78 0.74 0.77 0.78
Au3–SCH3 2.73 2.90 3.03 2.66 2.79 2.81 2.82 2.88
Au4–SH 2.18 2.36 2.49 2.09 2.23 2.25 2.27 2.48
Au+4–N2 0.67 0.80 0.92 0.58 0.68 0.70 0.72 0.74
Au4–SHCH3 0.95 1.11 1.24 0.93 1.04 1.06 1.07 1.05
ME -0.16 0.01 0.14 -0.28 -0.16 -0.13 -0.11
MAE 0.16 0.12 0.16 0.28 0.16 0.14 0.12
MARE 10.61% 8.84% 13.08% 18.89% 10.30% 9.16% 7.72%
StdDev 0.14 0.14 0.15 0.15 0.14 0.13 0.12
RESULTS some cases, e.g. for the K9 test where PBE0 and hP-
BEint with n = 4 perform best (see also the BH76RC
Quantum chemistry and solid-state benchmarks test [29, 47]), but has small effects for other cases, e.g.
G2RC, NBPRC and DC9 tests [47, 67]. Furthermore, in
mostsituationsimprovementsaresmallerorevenabsent
In Table I we report the mean absolute errors (MAE)
when large fractions of Hartree-Fock exchange are used;
fordifferenttests,asobtainedfromPBEintandhPBEint
see for example BHLYP vs BLYP and B3LYP for the
calculationswithn=3,4,5,withtheaimofassessingthe
BH76RC test [29, 47]. In particular, for the DC9 and
performance of the functionals for a broad set of prob-
OMRE benchmarks considered here, we find that PBE0
lems. To this end, also PBE, PBEsol, and PBE0 results
brings almost no improvement or even a worsening of
are reported, since these are natural references for the
the results with respect to PBE. Analogously, the best
functionals considered here. Full results of all tests are
results at the hPBEint level are obtained when n = 6
available in supporting information [58].
and a relatively small fraction of nonlocal exchange is
Foratomizationenergiesoforganicmolecules,itiswell
used.
knownthatGGAs arenotveryaccurate(PBEsollargely
fails), while the hybrid functionals perform better than Different conclusions are found when one considers
the corresponding GGAs and this is more so for those the energetic propertiesoftransition-metalcomplexesor
functionals including a larger fraction of Hartree-Fock clusters(TM10,AUnAE,TMRE).Inthiscaseinfactthe
exchange [29]. Consequently, PBE0, with a MAE of 5 use of PBE0 corresponds always to a considerable wors-
kcal/mol,isthebestfunctionaloverallandhPBEintwith ening of the results with respect to PBE.Aworseningof
n = 4 yields the best performance among the different theperformanceisalsoobservedinallcasesforhPBEint
variantsofhPBEint,reachingalmostthePBE0accuracy with respect to PBEint when n = 4, although not so
forthe W4test. Notethatevenbetter resultscanbeob- markedasforPBE0vsPBE.Improvementswithrespect
tainedby consideringhigherfractions ofexactexchange, to PBEint are instead found for hPBEint with n = 6,
as shown for example in Ref. 29 where the admixture of whichbecomesthe bestfunctionalforTM10,butnotfor
PBEGGAand32%ofexactexchangewasfoundtoyield the TMRE test. In this last case hPBEint with n=6 is
the best results for thermochemistry. definitelythe besthybridfunctionalinthepresentinves-
Whereas hybrid functionals improve significantly at- tigation, but slightly worst than PBEint and more than
omization energies, for reaction energies involving or- twiceworstthanPBE.NotehoweverthattheTMREtest
ganic molecules the use of hybrid functionals is not so is a particular difficult test in the context of the present
systematically beneficial [47, 67, 68]. In fact, inclusion work, because it requires a delicate balancing between
of Hartree-Fock exchange brings some improvements in the description of metal complexes (best described by
7
PBEintand PBE)andsmall organicmolecules (best de- a global MAE of about 0.28 Debye. A MAE larger by
scribed by hybrids with a large fraction of Hartree-Fock 30%isfoundinsteadforGGAmethodswhichperformall
exchange). verysimilarlyanddisplayaglobalmeanabsoluteerrorof
Concerning structural properties, hybrid functionals about 0.35 Debye. In particular, the best results are ob-
provide a modest improvement for organic molecules tainedwhenthelargerfractionofHartree-Fockexchange
(MGBL19 and F38), while they worsen the results for (0.25 for PBE0 and hPBEint with n=4) is considered,
gold clusters (AUnBL). Interestingly, the hPBEint func- having a MAE of 0.27 Debye and a MARE below 8%.
tionalwithn=6yieldsthebestresultsamongthediffer- Nevertheless, good results are achieved also by hPBEint
ent variants of hPBEint, outperforms PBE0 for AUnBL with n=6, including only a relatively small fraction of
and F38, and gives almost the same result as PBE0 for Hartree-Fock exchange.
MGBL19. Thus, the combination of PBEint with 1/6 of We note however that the results of Tab. II show for
Hartree-Fock exchange proves to be a very good choice all the functionals severalimportant deviations from the
for structural properties of finite systems. trends depicted above. In fact, for some systems (e.g.
For non-bonded interactions very good results are ob- HNCO, pyridine, ClCN) the best results are achieved at
tained at the GGA level using the PBE and PBEint the GGA level, while PBE0 yields the worst agreement
functionals, while poorer results (especially for hydro- with the reference. Thus, none of the functionals con-
gen bonds) are given by the PBEsol functional. Hybrid sidered here can be labeled as fully reliable for the cal-
functionals cannotimprovethe performance ofPBE and culation of the dipole moment of an arbitrary molecular
PBEint,yieldingthesameMAEastheGGAsfortheDI6 system, but the best compromises are provided by the
test and slightly larger errors for the HB6 test. In this hPBEint functional with n=6, which yields the largest
latter case, the best results at the hybrid level are ob- number of results with errors ≤0.1 Debye, and PBE0
tained by PBE0, taking advantage of the high accuracy that yields the best overall MAE and MARE and the
of PBE for the hydrogen bonds, and by hPBEint with largest number of results with errors ≤0.2 Debye.
n=6, taking advantage of the small fraction of Hartree-
Fock exchange included.
Concerning solid-state properties, PBEsol is the most Metal-molecule interfaces
accurate for lattice constants, while PBEint is interme-
diated between it and PBE. However, the limited accu- Inthissubsectionweinvestigatetheabilityofdifferent
racy of PBEsol for atomic energies causes it to perform functionals to describe hybrid metal-molecule interfaces.
poorly for cohesive energies (COH6 test). In this case, These systems pose a difficult challenge to any compu-
whereagooddescriptionofbothbulksolidsandatomsis tational method because of the very different theoreti-
required,PBEintshowsaremarkablygoodperformance, cal and numerical issues raised by metallic (extended)
due to its well balanced description of different density systems and molecules. The former are in fact mainly
regimes. characterized by a slowly-varying density regime and
can be properly described with local or semilocal ap-
proaches. The latter instead display a significant contri-
Dipole moments bution from rapidly-varying density regions and require
a proper treatment of nonlocal interactions.
In Tab. II we report the performance of the different In view of practical applications it would be nice, of
functionals for the description of dipole moments of sev- course, to perform tests for the interaction of (function-
eral test molecules. Benchmarking the dipole moments alized) organic molecules with relatively large metallic
is not as common as energy and structural tests, never- clusters or extended (eventually semi-infinite) surfaces.
theless it is very important to verify the accuracy in the However,forthesesystemsnobenchmarkvaluetoassess
descriptionofdensity [69,70]. This test allowsin fact to the performance of the methods can be obtained. Accu-
probe indirectly the quality of the description, provided rateexperimentalvaluesareinfactextremelyrare,while
by different functionals, of the electron density distribu- high level approaches cannot be applied due to their ex-
tion in molecules, especially in valence and asymptotic ceedingly high computational cost. Note also that for
regions. ThisissueisparticularlyrelevantforGGAfunc- large metallic systems static and high-order correlation
tionals, because due to the Coulomb self-interaction er- effects start to play a fundamental role, thus accurate
ror, often they yield a poor description of the electron results can only be obtained through very sophisticated
density far from the atomic core. For the same reasons, methods.
an improved description is usually obtained by hybrid For the reasons mentioned above, in this paper we re-
approaches. strict our attention to a set of relatively small metal-
The general trends outlined above are confirmed by molecule systems composed of small organic molecules
inspection of the results of Tab. II. The smallest er- interacting with gold clusters of 2, 3 and 4 atoms. For
rors are obtained by the hybrid functionals, which yield suchsystemsquiteaccuratebenchmarkresultshavebeen
8
TABLE IV:Mean absolute errors (MAE) in Hartreeor mean absolute relative errors (MARE) for different functionals for the
proposed model systems. Best results are in bold style; worst results are underlined.
System PBE PBEint PBEsol PBE0 hPBEint
n=4 n=5 n=6
Jellium surf. energy (MARE) 3.15 2.29 2.32 1.88 2.29 2.29 2.29
Jellium spheres (MAE) 0.46 0.46 0.56 0.57 0.23 0.18 0.17
Hooke’s atom (MARE) 3.63 3.45 3.41 3.87 3.78 3.71 3.66
Fractional spin H (MAE) 29.59 27.78 27.63 42.31 40.80 38.19 36.46
TABLE V: XC jellium surface energies (erg/cm2) of the global hybrids PBE0 and hPBEint. The fixed-node diffusion Monte
Carlo (DMC) calculations [73, 74] are reported for reference. (1hartree/bohr2 = 1.557×106erg/cm2.). PBE, PBEint, and
PBEsol results are taken form Table I of Ref. 24.
r PBE PBEint PBEsol PBE0 hPBEint DMC
s
n=4 n=5 n=6
2 3265 3378 3374 3312 3378 3378 3378 3392 ± 50
3 741 774 774 756 774 774 774 768 ± 10
4 252 267 267 259 267 267 267 261 ± 8
6 52 56 56 55 56 56 56 53 ±...
obtained with coupled cluster single, doubles with per- MAEandMAREofallthefunctionalsconsideredinthis
turbative triple correction (CCSD(T)) calculations and work. We note also that, because of the small fraction
an assessment of DFT methods could be carried out. ofHartree-Fockexchangethat itincludes, hPBEintwith
Results are reported in Tab. III. At the GGA level, n = 6 may be supposed to be the best compromise for
PBE presents a general tendency to underestimate the interfacesofinterestinpracticalapplication,wherelarge
interaction energies, while a slight overestimation is ob- metallic systems are involved [71, 72].
served by PBEint. Finally, PBEsol shows a rather
strong overestimation of the interaction energies. Over-
all, PBEint (with a MAE of 0.16 kcal/mol) is slightly Model systems
better than PBE and PBEsol. In addition, we note that
most of the accuracy for PBE comes from the smallest In this subsection, we show the performance of PBE0
systems(e.g.,Au –SH)andlargererrorsarefoundwhen and hPBEint for several important model-systems:
2
the system size increases. For PBEint an opposite be- (i) jellium surfaces, that contain the main physics of
havior is observed. This trend was already observed for simple metal real surfaces;
gold clusters [22] and led to the conclusion that PBEint (ii) jellium spheres, that are simple models for simple-
provides the most accurate description of medium and metal-particles;
largemetalclusters. A similarextrapolationcanbe sup- (iii)Hooke’satom,thatrepresentstwointeractingelec-
posedtoapplyalsointhe presentcaseofmetal-molecule trons in an isotropic harmonic potential of frequency ω.
interfaces, suggesting that superior results can probably Atsmallvaluesofω,theelectronsarestronglycorrelated,
be achieved by PBEint for realistic interfaces. This con- and at large values of ω, they are tightly bounded, two
clusion is also supported by the observation that for in- important cases in many condensed matter applications.
terfaces involving large systems slowly-varying density (iv) The hydrogen atom with fractional spin, a model
regions will gain a more important role. Thus PBEint for the static correlation given by degenerate states.
(and partially PBEsol) will be favored over PBE. For allthese cases,we use non-self-consistentaccurate
TheadditionofafractionofHartree-Fockexchangein calculations (for jellium surfaces we use numerical LDA
the functionals has the effect to reduce the computedin- orbitalsanddensities,forjelliumspheresweuseaccurate
teraction energies. This results in a worsening of PBE0 exact exchange orbitals and densities, for the Hooke’s
with respect to PBE, but an improvement of hPBEint atomweuseexactorbitalsanddensities,andforthespin-
with respect to PBEint. In fact, all the variants of hP- dependenthydrogenatomweuseexact-exchangeorbitals
BEint show a MARE lower than PBE and PBEint. In and densities.) For the hybrids, we use exact-exchange
particular, the option with n = 6 displays the smallest instead of the Hartree-Fock one.
9
The mean absolute (relative) errors of different func- 0.001
tionals for the selected model problems are summarized
in Tab. IV. A detailed analysis of the results is given in
the following.
0
Jellium surfaces E
∆
-0.001
PBE
The PBEint GGA performs remarkably well for the PBE0
PBEint
jellium surfaces, giving very accurate exchange and XC PBEint0 (n=4)
surfaceenergies[24,26]. Becausetherearenoerrorcom- PBEint0 (n=5)
-0.002 PBEint0 (n=6)
pensations between the exchange and correlation parts
and the PBEint exchange surface energies are extremely
0 20 40 60 80 100
closetotheexact-exchangeones[26],anyPBEinthybrid N
will yield the same results as the PBEint GGA and thus
be veryaccuratefor the jellium surfaces,aswe explicitly FIG. 1: Errors in total energies per electron (Eexact/N −
showintheTableV.Thisisaveryimportantresultfora Eapprox/N; Hartree) of jellium spheres for rs =4. Eexact/N
functional designed for hybrid interfaces, where the sur- representsthecorrectedDMCdata[77]. Weusemagicneutral
− − − − − − −
clusters with N=2e , 8e , 18e , 20e , 34e , 40e , 58e ,
facephysicsplaysadominantrole. Wealsomentionthat
− −
92e , and 106e .
PBE0improvesconsiderablyoverPBE,butstillitisnot
so accurate in the range 2 ≤ r ≤ 3, where important
s
metals lie (e.g. Al has rs =2.07, and Cu has rs =2.67). alsoa veryimportantresultfor a functionaldesignedfor
We mention that the RGE2 GGA of Ref. [75] gives hybrid interfaces, where metal particles play a crucial
similar jellium xc and x-only surface energies as PBEint role.
(seeTable3andFig. 3ofRef. [75]foradetailedanalysis
ofjelliumsurfaces). BothRGE2andPBEinthavetheex-
changeenhancementfactorF →1+µs2+O(s6)atsmall Hooke’s atom
x
s, and thus the requirement of vanishing O(s4) terms in
this expansionseemsveryimportantforaccuratejellium In Fig. 2, we report the relative absolute errors of the
x-only surface energies at the GGA level. Note that the hybrids,fortheHooke’satomwithseveralfrequenciesω.
exchange part of revTPSS meta-GGA [76], which recov- Inthetightly-boundedregime(r =(ω2/2)−1/3 issmall)
0
ers the 4-th order gradient expansion of the exchange all the functionals performs similarly, though PBE0 is
energy, is also accurate for jellium surfaces. the best in this extreme case. On the other hand, for
r > 5, that is the physical case for most of the real ap-
0
plications, the hPBEint hybrids are always better than
Jellium spheres PBE0, with a superior performance for the n = 6 case.
However,the errorsincrease when the strong correlation
In Fig. 1, we show the errors in total energies per increases, showing that these global hybrids can not de-
electron (Eexact/N −Eapprox/N) versus the number of scribe satisfactorily the strongly correlated regime.
electronsN,forjelliumspheresofmagicnumbers(N=2,
8, 18, 20, 34, 40, 58, 92, and 106) with bulk parameter
r = 4. (Note that sodium has r = 3.93.) The refer- Hydrogen atom with fractional spin
s s
ence values Eexact/N representthe correctedDMC data
(See Eq. (40) and Fig. 5 of Ref. 77). We note that Forthestaticcorrelationcase,theexactexchangefails
HF and EXX perform similarly and badly for this prob- badly, and thus the global hybrids usually increase the
lem [78], whereas LDA is remarkably accurate for large error in the static correlation with respect to the orig-
clusters [79, 80]. We also recall that self-consistent ef- inal GGA. So, from this point of view, the HF mixing
fects are very small (see Table I of [81]), such that our should be as small as possible. In Fig. 3, we show that
non-self-consistent results are expected to give an hon- indeed for the different variants of hPBEint increasingly
est and accurate picture of the performance of various good results are obtained in the series n=4, 5, 6. Thus,
functionals for jellium (metal) particles. hPBEintwithn=6has the bestperformance,while the
We observe that PBE0 does not improve upon PBE, worst results are obtained with PBE0, which is slightly
being even worse for N ≤ 34. On the other hand, hP- worst than hPBEint with n=4.
BEint(withn=4,5and6)significantlyimproveoverthe We recall however that the N-electron self-interaction
original PBEint for N ≤ 20. However, on average the error [82], or the delocalizationerror [83], that is related
best performance is given by hPBEint (n=6). This is to the convexity behavior of the functional at fractional
10
0.03 sults, the benchmarks were divided in two subsets: the
PBE
molecular-based tests including organic molecules, non-
PBE0
PBEint bonded interactions, transition metals and dipoles and
PBEint0 (n=4) the other-systems including interfaces, solid-state and
0.02 PBEint0 (n=5)
model systems. For the two groups RMAE-mol and
PBEint0 (n=6)
RMAE-other indicate respectively the average RMAE.
Exc Finally,RMAE-totindicatestheaveragedRMAEamong
∆
all seven benchmarks.
0.01
AmongtheconsideredGGAs,PBEintshowsaremark-
ably good performance yielding the best global RMAE-
otherandRMAE-totandaRMAE-molnotmuchhigher
than PBE and even lower than PBE0. In fact, despite
0
PBEint provides the smallest MAE only in few cases, it
5 10 15 20
r has a more balanced performance on different tests. On
0
the contrary, PBE0 and PBE give very accurate results
FIG. 2: ∆E =|(Eapp−Eexact)/Eexact| (Hartree) for the in some cases but behave rather poorly for other tests.
xc xc xc xc
Hooke’s atom with different frequencies. r0 = (ω2/2)−1/3 is The good balance over a broad set of systems and prop-
theclassical electron distance [26]. erties is an important feature of the PBEint functional
which makes it suitable for applications in complex sys-
0.15 tems where different situations (or density regimes) may
coexist.
Concerning the hybrid functionals similar considera-
0.12 PBE
PBE0 tions apply. We observe that functionals incorporating
PBEint 25%ofHartree-Fockexchange(PBE0andhPBEintwith
PBEint0 (n=4)
0.09 PBEint0 (n=5) n = 4), despite working very well for energetic proper-
PBEint0 (n=6)
E EXX ties of organic molecules (see Table I), perform globally
∆
worst than their GGA counterparts and other hybrids
0.06
with a lower Hartree-Fock content, again because of im-
portant failures for some properties (e.g., structures and
0.03 reaction energies). The best overall results are obtained
with hPBEint using n = 6, which gives also a RMAE-
mol of 0.95, (better than PBEint and PBE0). Notably,
0
0 0.2 0.4 0.6 0.8 1 hPBEint n=6 has in fact a nicely uniform performance
ζ
for all the tests considered in the present work, being
never the worst one and showing a good accuracy for all
FIG. 3: ∆E = E[n,ζ]−E[n,ζ = 1] (Hartree) versus the systems and properties.
spin-polarization ζ.
Moreover, hPBEint with n = 6 and PBEint are the
only functionals that can describe rather accurately at
the same time energetic and structural properties (this
number of electrons,decreaseswhen the mixing parame-
holds also for bulk in the case of PBEint).
terofaglobalhybridincreases. Thus,theerrorsofstatic
correlationandofdelocalizationcannotbebothreduced
by a global hybrid.
CONCLUSIONS
Global assessment In this paper, we performed an extended study of the
PBEint global hybrids. We tested important energeti-
cal and structural properties of organic molecules and
To compare the MAE of different tests, we computed,
metal complexes, as well as metal-molecule interfaces.
for each group of tests, the MAE relative to PBEint de-
We showed that the hPBEint functional, with the opti-
fined as
mal mixing parameter n=6,maintains the well-balanced
MAEi(method) behavior of the PBEint GGA functional, improving the
RMAE(method)≡ , (10)
MAE (PBEint) overall accuracy, over a wide range of problems and can
Xi i
thusbe successfullyemployedincomplexstudies suchas
where i runs over a given set of tests. The RMAEs for the description of hybrid metal-molecule interfaces.
the different classes of tests considered in this work are In fact, the PBEint functional and its hybrid forms
collected in Tab. VI. To help the analysis of the re- takebenefit fromthe goodtreatmentofboththe slowly-